In Virginia, early results showed a county official with a big lead over a former congresswoman in the bitter Democratic contest for the seat being vacated by retiring Republican U.S. Rep. Tom Davis.
And in Maine, Democrats chose between Rep. Tom Allen and political newcomer Tom Ledue in a U.S. Senate primary while both parties settled hard-fought races in the 1st Congressional District in the southern part of the state.
Graham challenger Buddy Witherspoon, a retired orthodontist and former Republican National Committee member, has criticized the one-term incumbent as too liberal for conservative South Carolina.
Graham has also taken heat from potential Democratic challengers on a failed illegal immigration measure he and McCain crafted.
Attorney Michael Cone and engineer Bob Conley were seeking the Democratic Party's nod to run for Graham's seat Tuesday. Graham's immigration measure, which would have provided a path to citizenship, drew derision last year at the state GOP convention, and Cone and Conley both call the plan amnesty for illegal immigrants.
South Carolina resident Jamie Devane said his vote for Witherspoon Tuesday was a vote against Graham on the immigration issue.
"I just think it's time to kick the people out who aren't supposed to be here," he said.
But Graham, an Air Force Reserve colonel who has served in Iraq, has name recognition and a huge financial advantage. McCain won the January primary in South Carolina and Graham has used his endorsement in television ads.
"I think he votes in our best interest, to take care of our troops and needing an actual immigration law instead of the way it is right now," said retired construction worker Glenn Muskovin, who said he voted for Graham even though he doesn't always agree with him. "He hasn't wavered like a lot of politicians."
Farther north, Democrats are hoping to pick up a seat in Virginia's 11 Congressional District, where Republican moderate Davis is retiring after nearly 14 years. He represents suburbs in and around the Capital Beltway, including much of Fairfax County, the wealthiest in the nation.
The two leading Democratic candidates are longtime rivals. Gerry Connolly is chairman of the Fairfax County Board of Supervisors and former Rep. Leslie Byrne held the seat for one term before losing it to Davis in the Republicans' 1994 takeover of Congress. With 30 percent of precincts reporting, Connolly had 57 percent of the vote, compared with 33 percent for Byrne. Two other Democrats trailed.
The primary winner will face Republican Keith Fimian, a businessman making his first run for office.
Voters in northern Virginia were also deciding nomination fights in two other congressional districts Tuesday.
In Maine, Allen, a six-term congressman from Portland, was favored to defeat Ledue, a high school administrator from Springvale, for the chance to challenge two-term Sen. Susan Collins. She was unopposed in the Republican primary.
Southern Maine's 1st District, where Allen is giving up his seat, featured a hotly contested six-way Democratic race and a two-way Republican contest.
And in Georgia, a political consultant and a national guardsman serving in Iraq were in a runoff to replace a state lawmaker facing prison time for money laundering. The House seat representing an area east of Atlanta was vacated by state Rep. Ron Sailor, who pleaded guilty to laundering $375,000 in drug money for an undercover agent posing as a drug dealer.
